Question

\[\int\frac{1}{\left( x^2 - 1 \right) \sqrt{x^2 + 1}} \text{ dx }\]

Answer 1

\[\text{ We  have,} \]
\[I = \int \frac{dx}{\left( x^2 - 1 \right) \sqrt{x^2 + 1}}\]
\[\text{ Putting  x }= \frac{1}{t}\]
\[ \Rightarrow dx = - \frac{1}{t^2}dt\]
\[ \therefore I = \int \frac{- \frac{1}{t^2}dt}{\left( \frac{1}{t^2} - 1 \right) \sqrt{\frac{1}{t^2} + 1}}\]
\[ = \int \frac{- \frac{1}{t^2} dt}{\frac{\left( 1 - t^2 \right)}{t^2} \times \frac{\sqrt{1 + t^2}}{t}}\]
\[ = \int \frac{- t \text{ dt }}{\left( 1 - t^2 \right) \sqrt{1 + t^2}}\]
\[\text{ Putting  1 }+ t^2 = u^2 \]
\[ \Rightarrow t^2 = u^2 - 1\]
\[ \Rightarrow 2t\text{ dt }= 2u \text{ du }\]
\[ \Rightarrow t \text{ dt } = u \text{ du }\]
\[I = - \int\frac{u \text{ du}}{\left( 1 - u^2 + 1 \right)u}\]
\[ = - \int \frac{du}{2 - u^2}\]
\[ = - \int \frac{du}{\left( \sqrt{2} \right)^2 - u^2}\]
\[ = - \frac{1}{2\sqrt{2}}\text{ log} \left| \frac{u + \sqrt{2}}{u - \sqrt{2}} \right| + C\]
\[ = - \frac{1}{2\sqrt{2}}\text{ log }\left| \frac{\sqrt{1 + t^2} + \sqrt{2}}{\sqrt{1 + t^2} - \sqrt{2}} \right| + C\]
\[ = - \frac{1}{2\sqrt{2}}\text{ log} \left| \frac{\sqrt{1 + \frac{1}{x^2}} + \sqrt{2}}{\sqrt{1 + \frac{1}{x^2}} - \sqrt{2}} \right| + C\]
\[ = - \frac{1}{2\sqrt{2}}\text{ log} \left| \frac{\sqrt{x^2 + 1} + \sqrt{2}x}{\sqrt{x^2 + 1} - \sqrt{2}x} \right| + C\]